Classification theory for abstract elementary classes

Rami Grossberg

ABSTRACT. In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the first-order case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah’s categoricity con-

jecture for Lω1,ω. While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galois-stability.

Introduction In recent years the view that stability theory has wider applicability than the originally limited context (i.e. first-order stable theories) is getting increasing recognition among model theorists. The current interest in simple (first-order) the- ories and beyond signifies a shift in the opinion of many that similar tools and concepts to those of basic stability theory can be developed and are relevant in a wider context. Much of Shelah’s effort in model theory in the last 18-19 years is directed toward development of classification theory for non elementary classes.I feel that the study of classification theory for non elementary classes will not only provide us with a better understanding of classical stability (and simplicity) theory but also will develop new tools and concepts that will be useful in projecting new light on classical problems of “main stream” mathematics. The purpose of this article is to present some of the basics of classification theory for non elementary classes. For several reasons I will concentrate in what I consider to be the most important and challenging framework: Abstract Elemen- tary Classes, however this is not the only important framework for such a classifi- cation theory.

1991 Mathematics Subject Classification. Primary: 03C45, 03C52, 03C75. Secondary: 03C05, 03C55 and 03C95. I would like to express my gratitude to the anonymous referee for a most helpful report and many valuable comments. Also I am grateful to Yi Zhang, without his endless efforts this paper would not exist. 1 2 RAMI GROSSBERG

I made an effort to keep most technical details to a minimum, while still having some deep ideas of the subject explained and part of the overall picture described. In the last section I discuss direction for future developments and some open prob- lems. An interested reader can find more details in Chapter 13 of [Gr]. I thank Oleg Belegradek for an interesting discussion and suggesting several examples, Andres´ Villaveces for his comments and Monica VanDieren for com- ments on a preliminary version.

1. definitions and examples

DEFINITION 1.1. Let K be a class of structures all of the same similarity type L(K). K is an Elementary class if there exists a first-order T in L(K) such that K =Mod(T ). There are very many natural classes that are not elementary: Archimedean ordered fields, locally-finite groups, well-ordered sets, Noetherian rings and the class of algebraically closed fields with infinite transcendence degree. Extensions of predicate calculus permit a model-theoretic treatment of the above: 1. The basic infinitary languages: Lω1,ω L+,ω L+, and L∞,ω. The earliest work on infinitary logic was published in 1931 by Ernst Zermelo [Z]. Later pioneering work was done by: Novikoff (1939, 1943), Bochvar (1940) ([Bo]) and from the late forties to the sixties the main contributors were: Tarski, Hanf, Erdos,˝ Henkin, Chang, Scott, Karp, Lopez-Escobar, Morley, Makkai, Kueker and Keisler. Recall DEFINITION 1.2. Lω1,ω contains all the first-order formulas in the language L and is closed under propositional connectives, first-order quantification and the rule: If {ϕ (x) | n<ω}L then n _ ω1,ω ∈ ϕn(x) Lω1,ω,`(x) <ω. n<ω

If {ϕ(x) | <}L + then _ ,ω ϕ(x) ∈ L+,ω. <

If {ϕ(x) | <}L + with `(x) <then _ , ϕ(x) ∈ L+, < CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 3

and for every sequence of variables hx | <<i

Qx0Qx1 Qx ϕ(hx | <i) ∈ L+, for Q ∈{∀, ∃}. For a limit cardinal : [ L, := L+,. <

L∞, is the proper class obtained by letting vary over all cardinals. I.e. it is [ L∞, := L+,. ∈Card

Basic references: For Lω1,ω see H. J. Keisler [Ke2] and for L+, see M. A. Dickmann’s [Di] books. 2. Cardinality quantifiers: Andrzej Mostowski in 1957 (see [Mos]) introduced several cardinality quantifiers. The most popular among them was studied extensively in the sixties by Gerhard Fuhrken [Fur] and Jerry Keisler [Ke1]. ℵ It is the 1-interpretation: L(Q) and extensions like Lω1,ω(Q) where M |= Qxϕ(x) ⇐⇒ { a ∈|M| : M |= ϕ[a]} is uncountable. 3. Other second order quantifiers: Measure-theoretic quantifiers, and topolog- ical logics. Barwise, Makkai, Kaufmann, Flum, Ebbinghaus and Ziegler. 4. Other logics: Monadic logics. Rabin, Gurevich, Buchi,¨ Shelah. There are many more logics, see the volume edited by Jon Barwise and Solomon Feferman [BaFe]. There is a particularly rich model theory for L(Q). In the last 20 years, this theory took on a set-theoretic flavor, see: Fuhrken [Fur], Keisler [Ke1], [Sh 82], [Gr2] and [HLSh]. For an overview of some of this I recommend Wilfrid Hodges’s book [Ho1]. A common feature of all the above extensions of first order logic is the failure of the compactness theorem. An ordered field hF, +, , i is archimedean iff _ hF, +, , i |= ∀x x>0 → [|x + ...{z + x} 1] . n<ω n-times A group G is periodic iff _ G |= ∀x [xn =1]. n<ω

DEFINITION 1.3. Let M and N both be L-structures. Suppose that A |M|, and B |N|. A function f : A → B is called a partial isomorphism iffitisa bijection and for every relation symbol R(x) and every a ∈ A we have that a ∈ RM ⇐⇒ f(a) ∈ RN , for every function symbol F (x) we have that 4 RAMI GROSSBERG f(F M (a)) = F N (f(a)) and for every constant symbol c if cM ∈ Dom(f) then f(cM )=cN . A family F of partial isomorphisms from M into N has the back and forth property iff (forward) for every a ∈|M| and every g ∈Fthere exists h ∈Fsuch that g h and a ∈ Dom(h) and (back) for every b ∈|N| for every g ∈Fthere exists h ∈Fsuch that g h and b ∈ Rang(h). Denote this by F : M =p N . We write M =p N for “there exists a non empty F such that F : M =p N”. At first glance the following special case of a theorem of Carol Karp may look a bit surprising:

FACT 1.4 (Karp’s test). Let M and N be L-structures. p M = N ⇐⇒ M ∞,ω N. Shelah’s plenary talk at the International Congress of Mathematicians in Berke- ley in 1986 (see [Sh 299] and [Sh tape]) was dedicated to classification theory for non elementary classes and in particular universal classes. There are several frameworks for classification theory for non elementary classes. With the exception of the last item they are listed in (more or less) increasing level of generality: Homogeneous model theory (this was formerly called Finite diagrams sta- ble in power). In this context we have a fixed (sequentially) homogeneous monster model M and limit the study to elementary submodels of M and its subsets. The subject was started by Shelah in 1970 with [Sh3] and con- tinued by him with [Sh54]. In the last 10-15 years the subject was revived in several publications. See [Gr3], [Gr4],[HySh1], [HySh2], [GrLe1], [GrLe3], [Le1], [Le2], [BuLe], [Be] and [Kov1]. Submodels of a given structure. This is a generalization of homogeneous model theory. Start with a given model M (not necessarily homogeneous) and limit the study to submodels of M and its subsets. Started in Grossberg [Gr3], [Gr4]. Further progress is in [GrLe2]. Excellent classes. An excellent class consists of the atomic models of a first- order countable theory which satisfy a very strong amalgamation property: The (ℵ0,n)-goodness for every n<ω. Shelah introduced them in [Sh87a], ∈ [Sh87b] as a tool to analyze Mod() when Lω1,ω under the model the- ℵ oretic assumption that I(ℵn,) < 2 n holds for every positive integer n. Later in [Sh c] Shelah used excellent classes at a crucial point in his proof of the main gap for first-order theories, (ℵ0,n)-goodness is renamed in [Sh c], page 616 as (ℵ0,n)-existence property. In section 5 of chapter XII She- lah essentially show that for countable and superstable T without the dop, notop is equivalent to the (ℵ0, 2)-existence property which using his resolu- tion technique from [Sh87b] implies excellence. Lately Kolesnikov [Kov2] announced an n-dimensional analysis to get new results for simple unstable CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 5

theories. Grossberg and Hart in [GrHa] developed orthogonality calculus to the level that permitted deriving the main gap for excellent classes. This was the first time that a main gap was proved for non elementary classes. Universal classes. TheV prototypical example is Mod() when is an Lω1,ω sentence of the form n<ω n where the n are 1-first-order sentences. This work began in [Sh 300]. Shelah is currently writing a book [Sh h] that, among other things, will include a “main gap”-style of theorem for universal classes. Abstract elementary classes. See Definition 1.5 below. This is in my opin- ion the deepest direction. It is the focal point of this article. Already in the fifties model theorists studied non elementary classes of structures (e.g. Jonsson´ [Jo1], [Jo2] and Fra¨isse[´ Fre]). In [Sh88], Shelah introduced the framework of abstract elementary classes and embarked on the ambi- tious program of developing a classification theory for abstract elementary classes. This work was continued in many publications of Shelah (totaling more than 700 pages) and members of his school. Primal framework. This is a generalization of abstract elementary classes obtained by relaxing the chain axioms (A4 from Definition 1.5 below). See Baldwin and Shelah’s papers [BS1], [BS2], [BS3] and [Gr5]. Classification theory over a predicate. Unlike the other frameworks this is really an extension of first-order model theory, when the notion of isomor- phism is replaced by a stronger one. While in my opinion this framework does not precisely fit into what I call classification theory for non elemen- tary classes, many of the methods are common. The fact that many years ago Shelah announced a solution for the main gap in this context, while for AECs such a theorem is not even on the horizon, indicates to me that this framework is much easier. I suggest to the reader to start with Wil- frid Hodges’s survey article [Ho2]. Pillay and Shelah’s article [PiSh]isthe beginning. Further work includes [Sh 234] and [Sh 322]. Shelah has writ- ten several hundreds of (unpublished) pages that continue this up to a Main gap. This work is not available to me. I suggest that interested people will contact Shelah directly. The focus of this article is the framework of abstract elementary classes. This framework, in my opinion, has the best balance of generality, a rich and sophisti- cated theory. The context of AECs is much more general than that of homogeneous model theory, model theory for Lω1,ω or even the framework of submodels of a given structure.

DEFINITION 1.5. Let K be a class of structures all in the same similarity type L(K), and let K be a partial order on K. The ordered pair hK, Ki is an abstract elementary class, AEC for short iff A0 (Closure under isomorphism) (a) For every M ∈Kand every L(K)-structure N if M = N then N ∈K. 6 RAMI GROSSBERG (b) Let N1,N2 ∈Kand M1,M2 ∈Ksuch that there exist fl : Nl = Ml (for l =1, 2) satisfying f1 f2 then N1 K N2 implies that M1 K M2. A1 For all M,N ∈Kif M K N then M N. A2 Let M,N,M be L(K)-structures. If M N, M K M and N K M then M K N. A3 (Downward Lowenheim-Skolem)¨ There exists a cardinal LS(K) ℵ0 + | L(K)| such that for every M ∈Kand for every A |M| there exists N ∈Ksuch that N K M, |N|A and kNk|A| + LS(K). A4 (Tarski-Vaught Chain) (a) For every regular cardinal and every N ∈Kif {Mi K N : Si<}KisSK-increasing (i.e. ⇒ ∈K i

EXAMPLE 1.7 (elementary classes). Let T be a first-order theory, K =Mod(T ) and K the usual notion of elementary submodel. Then hK Ki is an AEC with LS(K)=| L(T )| + ℵ0.

EXAMPLE 1.8 (ℵ1-saturated models of a f.o. theory). Let T be a complete count- able superstable and not ℵ0-stable, K be the elementary submodel relation and ℵ LS(K)=2 0 . K := {M |= T : M is ℵ1-saturated}. By Theorem III.3.12 of [Sh c] K is closed under unions (recall that superstability implies (T )=ℵ0). ∈ Recall (Keisler [Ke2]): for Lω1,ω a subset LA of Lω1,ω is a fragment containing iff ∈ LA, LA is closed under: Taking subformulas, first-order connectives and quantifiers.

DEFINITION 1.9. Let M and N be L-structures. Suppose LA is an L-fragment. M TV,LA N iff 1. M N and 2. for every a ∈|M| and every ϕ(y; x) ∈ LA if N |= ∃yϕ(y; a) then there exists b ∈|M| such that N |= ϕ[b; a]. When LA consists of only the first-order formulas from the language of the struc- ture we omit it. It is the contents of the Tarski-Vaught test that M is an elementary submodel of N iff M TV N. ∈ EXAMPLE 1.10 (Lω1,ω). Let Lω1,ω be a sentence in a countable language and suppose that LA Lω1,ω is a countable fragment containing . Take K to ⇐⇒ be defined by M K N M TV,LA N. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 7

Let K := Mod(T ). Clearly hK, Ki is an abstract elementary class with LS(K)=ℵ0. EXAMPLE 1.11 (Lω1,ω(Q)). T Lω1,ω(Q) be a countable theory in a count- able language and suppose that LA Lω1,ω(Q) is a countable fragment contain- ing T . hK, Ki is an abstract elementary class, when K =Mod(T ) and M K N iff M TV,LA N 1. if a ∈|M| and M |= Qxϕ(x; a) then there exists b ∈|N||M| such that N |= ϕ[b; a] and 2. if a ∈|M| and M |= ¬Qxϕ(x; a) then ϕ(M; a)=ϕ(N; a).

In fact AECs are more general than Lω1,ω(Q). The contents of the following Theorem is that in the Chain Axioms (A4)itis possible to replace the regular cardinal by an arbitrary directed set.

THEOREM 1.12 (Theorem 21.4S of [Gra]). Let hK, Ki be an AEC and hMs : ∈ i ∈K s I be a directed system. Then s∈I MSs . Moreover ∈ (a) If Ms KSN for every s I, then s∈I Ms K N. ∈ (b) Ms K t∈I Mt, for every s I. PROOF. Show directly for finite and countable I. For uncountable I by induc- tion on |I| using the following:

FACT 1.13. For every I uncountable directed set. There exists {I | <|I|} increasingS such that each I is a directed subset of I of cardinality || + ℵ0 and I = <|I| I.

An early version of 1.12 can be found in Tsurane Iwamura’s paper from 1944 ([Iw]). NOTATION 1.14. Denote by K the class {M ∈K: kMk <} and by K < the class {M ∈K: kMk = }. I(, K) is the cardinality of K / =.

Some examples from “main stream mathematics”: EXAMPLE 1.15 (Normed fields). Let K := {hF, +, , ||i|F is an algebraically closed field }

hF1, +, , ||1iK hF2, +, , ||2i⇐⇒F1 F2, (∀a ∈ F1)[|a|1 = |a|2] and the value groups are the equal. EXAMPLE 1.16 (Local fields). Let hF, +, , ||ibe a non archimedean normed field (i.e. |a + b|Max{|a|, |b|} for all a, b ∈ F ). It is well known that ([Cass]) R := {a ∈ F : |a|1R} is a subring of F and I := {a ∈ F : |a| < 1R} is a maximal ideal of R. The field R/I is called the residue field of F . A field is local iff its residue field is finite. The class K := {hF, +, , ||i : F is local} is an AEC when hF1, +, , ||1iK hF2, +, , ||2i is as in the previous example. Another relation K is defined by dropping the requirement of equal value groups. 8 RAMI GROSSBERG

EXAMPLE 1.17 (Noetherian rings). Define R K S iff R is a subring of S and R ∞,ω S. Notice that by Karp’s test when R is Noetherian then also S is.

EXAMPLE 1.18 (Rings of finite dimension). Let K be a class of rings. Define 0 0 R K S iff R is a subring of S and if I ( J are ideals of R and I ( J are ideals of S such that 1. I0 ∩ R = I, 2. J 0 ∩ R = J and 3. there is no ideal of R strictly between I and J then there is no ideal of S strictly between I0 and J 0.

EXAMPLE 1.19 (coordinate rings). Let F be a finite field or the rationals. Sup- n pose p ∈ F [x1,... ,xn] is such that {a ∈ F | p(a)=0} is an irreducible va- riety. For an algebraically closed field K F let Kp be the coordinate ring of {a ∈ Kn | p(a)=0}. The following is an AEC:

Kp := {Kp | K is an algebraically closed field extending F }.

The relation K is the subring relation. Recently motivated by a problem in transcendental number theory, Boris Zilber discovered the following most interesting example:

1.1. Zilber’s Schanuel structures. Some of the most intractable problems of number theory involve trascendental numbers. E.g. it is conjectured but unknown that the number e+ is trascendental. Schanuel’s conjecture (see 1.20 below) is a far reaching conjecture that implies the above (using the identity ei = 1) and several other (very difficult) conjectures. Let K h i| e := F, +, , exp F is an algebraically closed field of characteristic zero, ∀x∀y[exp(x + y)=exp(x) exp(y)] Kpexp := hF, +, , expi∈Ke | ker(exp)=Z Zilber introduced the following class: Kexp := hF, +, , expi∈Kpexp | Satisfying EC, CC and SCH EC is the essentially the requirement that the class is existentially closed. CC stands for countable closure property - every analytic subset of F n of dimen- sion 0 is essentially countable. SCH stands for

CONJECTURE 1.20 (Schanuel). For every x1,... ,xn ∈ F if {x1,... ,xn} are linearly independent over Q then

tr.degF/Q{x1,... ,xn, exp(x1),... ,exp(xn)}n. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 9 ∈K For M,N exp the relation M Kexp N is defined as M N and

tr.degM/Q{x1,... ,xn} = tr.degN/Q{x1,... ,xn} for every n<ωand for every x1,... ,xn ∈ M. hK i Zilber noticed that exp, Kexp is axiomatizable by a sentence of Lω1,ω(Q) when the similarity type is that of L(Ke). Zilber managed to show using Fraisse’s-´ type of construction:

FACT 1.21 (Zilber). Kexp is categorical in ℵ1 Using difficult field arithmetic (using the theories of fractional ideals in number fields as well as Weil divisors and the normalization theorem) Zilber managed to prove the following:

FACT 1.22 (Zilber [Zi1]). Given algebraically closed fields L1,... ,Ln C and elements a1,... ,an ∈ C the multiplicative group of the field

Q(L1,... ,Ln,a1,... ,an) can be presented as A T L1 Ln , where A is a free abelian, T is the torsion subgroup (when n =0). Recently Zilber observed that the model-theoretic contents of Fact 1.22 is that Kexp is an excellent class (in the sense of Shelah [Sh87b]). Recall that an AEC K K is excellent if ℵ0 satisfies a strong amalgamation property. Using two results of Shelah that were discovered in the late seventies (in the course of setting the fundations for classification theory for non elementary classes).1 FACT 1.23 (Shelah [Sh87b]). An excellent class has arbitrarly large models. FACT 1.24 (Shelah [Sh87b]). Let K be excellent. If K is categorical in an uncountable cardinal then K is categorical in every uncountable cardinal. Combining the last two together with Fact 1.22 Zilber concluded ℵ COROLLARY 1.25 (Zilber). Kexp has a unique object of cardinality 2 0 . Thus in order to prove Schanuel’s conjecture it suffices to show that the function exp(x) defined on C (obtained from the categorical structure) is indeed ex.

Zilber also noticed that the structures of Kexp are not sequentially homoge- neous, thus this class does not fit into the framework of homogenous model theory (previously known as finite diagrams stable in power). Thus it is a more “mathe- matical” example than Marcus’s example of a categorical class that is not homo- geneous from [Ma]. This is also the first example known to me for a categorical AEC very different from the previous ones. I think that it is astonishing that model theoretic work that in the past was viewed by many to be detached from the main body of mathematics which de- pends heavily on methods of combinatorial set theory can have connections with

1At first Zilber rediscovered in [Zi2] a special case of Shelah’s 1.24. 10 RAMI GROSSBERG

”concrete” mathematics, especially transcedental number theory. I am confident that in the future many other concrete examples will be found.

2. what is the purpose of this? Develop a classification theory for non elementary classes for K =Mod(T ) when T is a complete Lω1,ω-theory (or maybe some other non first-order theory). We try to imitate the rich and productive theory for elementary classes. Eventually this will have applications in “mainstream” mathematics. What is classification theory? We want to be able to answer every question about K / =. Basic test questions:

1. Given LS(K) is K =6 ∅? 2. Does K =6 ∅ imply K+ =6 ∅? 3. Does I(+, K)=1imply that I(, K)=1? 4. What are the possible functions 7→ I(, K)? 5. Under what condition on K is it possible to find a nice dependence relation on subsets of every M ∈K? In pure model theory the Łos´ Conjecture (see [Lo]) was a major driving force:

CONJECTURE 2.1 (Łos´ conjecture 1954 and Morley 1965). Let T be a first- order theory. If there exists >|T | + ℵ0 such that I(, T )=1then I(, T )=1 holds for every >|T | + ℵ0. In 1965 Morley [Mo1] confirmed the conjecture for theories in a countable lan- guage. For his proof Morley discovered the notions of Morley rank, prime model over a set and implicitly strongly minimal formulas. Important progress was made earlier by Ehrenfeucht and Mostowski as well as by Vaught. At the end of his article [Mo1] Morley raised the question whether his categoricity theorem holds for theories in uncountable languages. Several people recognized it as an impor- tant problem. Fredrick Rowbottom [Ro] and J. P. Ressayre [Re] made important progress toward a complete solution. In 1970, in addition to building on earlier work, Shelah invented superstable theories, weakly minimal formulas and local rank to prove the conjecture for all first-order theories. There is a need for good test-questions to measure progress in classification theory for non-elementary classes. Around 1977 Shelah proposed a conjecture that would serve as a benchmark for progress of the theory and may serve as a guide for future developments:

CONJECTURE 2.2 (Shelah’s conjecture). Let T be a countable theory in Lω1,ω. i If there exists ω1 such that I(, T )=1then I(, T )=1holds for every i ω1 . There is a similar conjecture for AEC generalizing the above conjecture. See Conjecture 3.6 in the next section. Based on experience with the first-order version it is likely that any attempt to prove 2.2 will produce a rich and powerful machinery. Indeed the partial results obtained so far indicate that this is the case. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 11

A word about examples: Essentially all known examples for categorical classes are derived from first-order ones. Generic example: Let T be a countable first-order theory categorical in ℵ1. Suppose M |= T of cardinality ℵ1, pick a countable A subset of |M|. Let L0 := L(T ) ∪{P } when P is a new unary predicate. ^ _ T,A := T ∧∀x[P (x) → x = a] a∈A

By the proof of Morley’s theorem, ℵ1-categoricity of T implies that also Th(hM,aia∈A) is categorical in every ℵ1. This is an example of an

Lω1,ω-theory categorical in all uncountable cardinals. The only exceptions are the example of coordinate rings (Example 1.19) which is categorical in every uncountable cardinal, Marcus’s example and Ziber’s Shanuel structures. I think it is too early at this stage of the theory to formulate a Zilber-like con- jecture of the nature of all categorical AECs or even the categorical Lω1,ω-theories, but perhaps there is a classifcation of the classes categorical classes above the Hanf number. Conjecture 2.2 is very open. There are more than 500 published pages dedi- cated for partial results. Among them are: 1. Keisler (1971), using a two cardinal theorem that improves Vaught’s the- orem, has shown that under the additional assumption of the existence of sequentially homogeneous model the conjecture is true. Unfortunately, Shelah observed that using a construction of Leo Marcus [Ma], Keisler’s additional assumption does not follow from categoricity. Thus Keisler’s strategy, while being very elegant, is a dead end. 2. Shelah (1978) in [Sh87a] and [Sh87b] building on [Sh48] proved a form ℵ of the conjecture under the additional assumption of I(ℵn+1,T) < 2 n+1 for every n<ω.Such a class of structures is called an excellent class. Grossberg and Hart [GrHa] proved the main gap for excellent classes. 3. Lessmann ([Le1]) proved the conjecture for countable finite diagrams, us- ing a Baldwin-Lachlan style argument by introducing the necessary prege- ometries via a new rank function. 4. Makkai and Shelah [Sh285] proved a downward version of the conjecture under the additional assumption that both and are above a strongly compact cardinal, and = +. It is a major open problem of [Sh 702]to get rid of the assumption that is a successor. There are nicely behaved forking and orthogonality calculi for this. 5. Kolman-Shelah [KoSh] and Shelah [Sh472] contain partial going down re- sults for above a measurable cardinal with the additional assumptions that K has the amalgamation property and is a successor cardinal. 6. [Sh 394] deals with classes that satisfy the amalgamation property. Sev- eral important concepts are introduced and a downward categoricity is con- cluded without using a large cardinal assumption as in [KoSh]. 12 RAMI GROSSBERG

7. In [ShVi] Shelah and Andres´ Villaveces embarked on an even more am- bitious program: Deal with the categoricity conjecture for classes with slightly weaker model and set-theoretic assumptions. Namely, they assume GCH and work with classes that have no maximal models (this is a weaken- ing of the amalgamation property). Recently more progress on this direction was made by Monica VanDieren [Va]. 8. Inspired by a question of mine, whether it is possible to generalize the re- sults of [Sh48] and [Sh88] for uncountable cardinals (or for PC classes for uncountable ). E.g. generalize [Sh48] from ℵ1 to arbitrary . the problem is that both in [Sh48] and [Sh88] the assumption that the categoricity is the successor of ℵ0 was used heavily in a form of applying a weak compactness

phenomenon (in the form of undefinability of a well ordering in Lω1,ω). I suspected that this attempt must produce new model-theoretic machinery. It turns out that I was right. In several massive papers Shelah answered that question and more. This appears in [Sh 576] (125 pages), [Sh 600] (82 pages), and [Sh 603] (20 pages). In particular he has shown (under weak GCH and no large cardinals) that if K is categorical in both and + then ++ ++ I( , K) < 2 =⇒K+++ =6 ∅. 9. Depending heavily on [Sh 576] Shelah in [Sh 705] is developing the analog theory of excellent classes (from [Sh87b]) for AECs. The current draft of the paper has more than 120 pages in it. In 1986 I proposed ∈ CONJECTURE 2.3 (Intermediate Łos´ conjectures). Let Lω1,ω i 1. If is categorical in some ω1 , then Mod() has the amalgamation i property in every ω1 . i 2. If Mod() has the amalgamation property for all ω1 , then categor- i i ical in some ω1 implies that categorical in for every ω1 . In section 6 it will be shown that the amalgamation property permits a nice theory of types. In [KoSh] Oren Kolman and Shelah derive the amalgamation property from the assumption that is categorical above a measurable cardinal. Lately Shelah and Villaveces in [ShVi] have shown that if K has no maximal models (i.e. every M ∈Khas a proper K-extension in K) the weak GCH implies that every model can be extended to an amalgamation base. [Sh 394] is dedicated to progress toward #2. It contains a proof of a downward version of the categoricity conjecture under the assumption that the class has the amalgamation property. WHY? What could be the benefits of such a theory? Looking at the first-order exam- ple the notions of independence, several model-theoretic rank functions, forking, orthogonality calculus, regular types, pre-weight, prime models etc. all have found concrete applications in algebra, generalizing Krull’s dimension theory from com- mutative algebra. 1. Clarify the above notions: CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 13

“Studying only the model theory of first-order logic would be analogous to the study of real analysis never knowing of any but the polynomial functions; core concepts like continuity, differ- entiability, analyticity, and their relations would remain at best vaguely perceived. It is only the study of more general functions that one sees the importance of these notions, and their different roles, even for the simple case.” -Jon Barwise, page 15–16 of [BaFe]. 2. It is beautiful and difficult mathematics. 3. Effect on model theory for first-order theories. Already [Sh87a] and [Sh87b] had a profound effect on the proof of main gap for first-order theories. Es- pecially good sets and stable systems. See the last 5 sections of Chapter XII in [Sh c] and what Shelah named the book’s main theorem-Theorem XII.6.1. I expect that a similar n-dimensional analysis will be used to better our understanding of simple unstable first-order theories. Also classification theory over a predicate benefited much from this. 4. Many interesting concepts of classical mathematics are not first-order (Archimedean, Noetherian and any chain conditions. etc). 5. Potential applications in classification theory of finite models. See Baldwin and Lessmann [BaLe] and Lessmann [Le3].

3. basic facts The following is the notion of pseudo-elementary (or projective) class.

DEFINITION 3.1. Let L1 L and let T1 be a first-order theory in L1, suppose that is a set of T1-types without parameters. We denote by

PC(T1, ,L)={ M L : M |= T1 and M omits all types from }. DEFINITION 3.2. A class K of structures is called a PC-class if there exists an expansion L1 of L(K), a first-order theory T1 in L1 and a set of T1-types such that K = PC(T1, ,L(K)). When |T1| + || + ℵ0 = we say that the class is PC. In the special case when L1 = L(K) we write EC(T,) for PC(T,,L(K)) and say that the class K is an EC-class.

THEOREM 3.3 (C.C. Chang 1968). If T is a theory in L+,ω of cardinality then Mod(T ) is PC. Similarly to Birkhoff’s presentation theorem for varieties/equational classes, there is a syntactic presentation theorem which generalizes Theorem 3.3:

THEOREM 3.4 (Shelah’s presentation theorem). LS(K) If (K, K) is an AEC, then there exists 2 such that K is a PC class. K K Also K := {(N,M) | M K N} is a PC class, for L(K ) consisting of a single unary predicate symbol. A proof uses Theorem 1.12, details can be found in [Sh88] or in Chapter 13 of [Gr]. 14 RAMI GROSSBERG

Using Morley’s result on the bound on computing Hanf numbers this imme- diately gives a corollary that is surprising and difficult to prove directly from the definitions, but is a trivial consequence of Shelah’s presentation theorem.

COROLLARY 3.5. Let (K, K) be an AEC. If K =6 ∅ for some i LS(K) (22 )+ then K =6 ∅ holds for all LS(K). A generalization of Conjecture 2.2 to AECs appears in [Sh 702]:

CONJECTURE 3.6 (Shelah’s conjecture for AEC). Let K be an AEC. If there i K K exists (2LS(K))+ such that I(, )=1then I(, )=1holds for every i (2LS(K))+ . Now we introduce the appropriate generalization of elementary embedding.

DEFINITION 3.7. For M,N ∈Ka monomorphism f : M → N is called a K K-embedding iff f[M] K N. Denote this by writing f : M ,→ N. When the identity of K is clear it is omitted and we write f : M,→ N.

DEFINITION 3.8. Let (K, K) be an AEC, and let , LS(K) be cardinals.

1. We say that a model M ∈K is a (, )-amalgamation base iff for every K M1 ∈K, M2 ∈K and f` : M ,→ M` for ` =1, 2, there exists N ∈Kso K K that there are g1 : M1 ,→ N and g2 : M2 ,→ N satisfying g1 f1 = g2 f2. Namely, the following diagram is commutative:

g1 / MO1 NO

f1 g2 / M M2 f2

N is called an amalgam of M,M1,M2,f1,f2. When = = we say that M is an amalgamation base. 2. M ∈K is an (<, )-amalgamation base iff for every 1 <, K ∈K ∈K → 1 , M1 1 , M2 1 and f` : M , M` for ` =1, 2, there exists an amalgam N of M,N1,N2,f1 and f2. 3. K satisfies the (, , )-amalgamation property iff every M ∈K is an (, )-amalgamation base. 4. K satisfies the -amalgamation property iff every M ∈K is an amalga- mation base. It is a corollary of the Robinson consistency theorem that if K =Mod(T ) for some complete first-order theory then K has the -amalgamation property for all ℵ0 + |L(T )|. K has the JEP iff for every M1,M2 ∈K there are N ∈Kand K-embeddings K f` : M` ,→ N. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 15

REMARK 3.9. Note that for first-order theories an even stronger amalgama- tion property holds. Every model is an amalgamation base for arbitrarily large models. I.e. remove the requirements that the models N` have the same cardinal- ity as M. The stronger amalgamation property is used in [Va], but we will work here with that from the previous definition (when both extensions are of the same cardinality as the base.) Already in 1956 B. Jonsson´ studied abstract elementary classes with the amal- gamation and JEP properties. This influenced Morley and Vaught to introduce saturated models ([MoV]). Shelah was the first in [Sh88] to consider AEC in the context of classification theory. Implicitly indicating that the methods of stabil- ity/classification theory are not limited to elementary classes only and potentially have a broader applicability than first order. In the proof of Morley’s categoricity theorem (as well as in Shelah’s general- ization to uncountable theories) saturated models play a central role. At first one shows that saturated models exists (at least at the categoricity cardinal ) and using the uniqueness of them it suffices to show that having an uncountable non-saturated model implies the existence of a non-saturated model in , contradicting categoric- ity. Non elementary classes in their very nature are connected to omitting types, so working with saturated models is not reasonable. It turns out that model homogeneity is a good replacement for saturation. In that it generalizes saturation (for elementary classes) and we have the analog existence and uniqueness theorems.

DEFINITION 3.10. Let K be an AEC. 1. Let >LS(K). We say that M is -model homogeneous iff for all N K 0 0 K N ∈K< such that N K M there exists f : N ,→ M such that f |N| = id|N|. 2. M is said to be model homogeneous iff M is kMk-model homogeneous. It is an exercise (using the compactness theorem) to show:

PROPOSITION 3.11. Let K be an elementary class. For >LS(K) and M ∈ K. M is saturated iff M is model homogeneous. By imitating the argument of the proof of existence of saturated models one can show:

THEOREM 3.12 (existence). Let K is an AEC and LS(K). Suppose that + 2 = . Further assume that K+ is not empty. If K has the -amalgamation property, then there exists a model homogeneous M ∈K+ .

THEOREM 3.13 (better existence). Suppose that K be an AEC such that > < LS(K) satisfies = and K is not empty. If K has the (<,)-amalgamation property, then for all N ∈K there exists M K N,M ∈K which is model ho- mogeneous. 16 RAMI GROSSBERG

THEOREM 3.14 (uniqueness). Let K be an AEC. Suppose that K is categori- cal for some LS(K).IfM and N are model homogeneous members of K+ , then M = N. There are uniqueness theorems that follow from weaker assumptions. While model homogeneity is very nice, model homogeneous models are nat- ural to consider only when K has the amalgamation property. I.e. Suppose that K is categorical in and K has no maximal models. The model of cardinality is model homogeneous iff K has the (<,)-amalgamation property for every <. A substitute called (, )-limit model was introduced in [KoSh] and where used in a substantial way to obtain the amalgamation property (from categoricity above a measurable cardinal), limit models reappeared also in [Sh 394] under the assumption that K has the amalgamation property. Further study of limit mod- els (without requiring the amalgamation property) is in Shelah and Villaveces in [ShVi]. The uniqueness: Any two (, )-limit models are isomorphic for different ’s was proved only lately by Monica VanDieren who has introduced and offered a characterization of the correct notion of model homogeneity for classes not requir- ing amalgamation in [Va]. Since most of this article deals with classes that have the amalgamation property I will not discuss (, )-limit models here.

4. the beginning of classification theory for AEC In his JSL list of open problems from 1975 Harvey Friedman reproduced a question that started classification theory for non elementary classes.

QUESTION 4.1 (Baldwin’s problem 1975). Does there exists a countable sim- ilarity type and a countable T L(Q) (in the ℵ1 interpretation) such that T has a unique uncountable model (up to isomorphism)? Since the Downward Lowenheim¨ Skolem theorem holds for L(Q), Baldwin’s question is equivalent to “Does there exists a countable similarity type and a count- able T L(Q) such that T is categorical in ℵ1 but does not have a model of cardinality ℵ2?” The question is important since it suggested for the first time a connection between categoricity in a cardinal and existence of models in its successor. A natural extension (generalizing L(Q) by an AEC and more importantly re- placing ℵ0 by an arbitrary ):

QUESTION 4.2. Let K be an AEC and LS(K). Does categoricity in + of K imply existence of a model of cardinality ++? The following is a relatively simple example of a family of deep results that was motivated by Baldwin’s question.

+ THEOREM 4.3. Suppose 2 < 2 . For an AEC K which fails to have the + -amalgamation property. If I(, K)=1and LS(K) then I(+, K)=2 . Section 8 of this paper is dedicated to the a proof of Theorem 4.3. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 17

+ The set-theoretic assumption 2 < 2 is known as the weak continuum hypothesis since it follows from 2 = +. However instead of using cardinal arithmetic we use a “diamond-like” combinatorial principle known as the Devlin- Shelah’s weak diamond: + FACT 4.4 (Devlin-Shelah). If 2 < 2 then there are + pairwise disjoint + 2 stationary subsets of such that for any of these sets S the principle + (S) 2 holds, where + (S) is: + For all F : >2 → 2 there exists g : + → 2 so that for every f : + → 2 the set { ∈ S | F (f )=g()} is stationary. For more details see [DS] or section 13.5 of [Gr] or Chapter 13 of [Sh f]. The use of the weak diamond is essential, the statement of Theorem 4.3 is false ℵ 0 ℵ under MAℵ1 +2 > 1:

FACT 4.5. There exists an AEC, K with LS(K)=ℵ0 such that 1. K is categorical in ℵ0, K 6 ∅ 2. ℵ1 = and K 3. The amalgamation property fails in ℵ0 . We have that ℵ 0 ℵ ⇒ ℵ K MAℵ1 +2 > 1 = I( 1, )=1. This class is obtained by essentially considering the countable substructures of the random bipartite graph whose left side is ω and right side is ω1. The categoricity proof is similar to Baumgartner’s ([Bau]) proof of the uniqueness of ℵ1-dense orders. I will conclude this section with a typical application of the weak diamond to AEC, this is not a particular case of Theorem 4.3 but rather a different theorem. The following theorem is a simple prototype of several more sophisticated results (e.g. [ShVi] and [Va]). A structure M ∈K is called universal model iff for every N ∈K there exists a K-embedding from N into M. + THEOREM 4.6. Suppose 2 < 2 . For an AEC K which fails to have the -amalgamation property. If I(, K)=1and LS(K) then K+ does not have a universal model. In this section, as well as in sections 5 and 8, I will use some elementary facts about stationary sets of ordinals (all can be found in Kunen’s book [Ku]or in section 1.8 of [Gr]): Let be an uncountable regular cardinal. A set C of ordinals all less than is called a closed unbounded set (club) iff for every < Sthere exists ∈ C such that >and for every bounded A C we have that A ∈ C. A set S is stationary iff S ∩ C =6 ∅ for every club C . FACT 4.7. Let be an uncountable regular cardinal. Let M be a structure of cardinality in a countable language. 1. If {Mi M | i<} and {Ni M | i<} areS elementaryS chains which are increasing and continuous such that M = i< Mi = i< Ni and kMik + kNik <, for all i<, then the set {<| N = M} contains a closed unbounded subset of . 18 RAMI GROSSBERG

2. Suppose that L(M) contains a unary predicate P such that P M is the set of ordinals less than . Then for every continuousS increasing elementary { | } k k chain Mi M i< such that M = i< Mi and Mi <for all i<the set {<| P M = } contains a club of .

2 Instead of using the principle + directly we use another combinatorial prin- ciple:

DEFINITION 4.8. + is said to hold if and only if for all + + + + {f : ∈ 2 where f : → } and for every club C , there exists + =6 ∈ 2 and there exists a ∈ C such that = , f = f and [] =6 [].

2 In [DS] Devlin and Shelah have shown that + follows from + (see also Chapter 13 of [Gr]). Now to the proof of Theorem 4.6:

PROOF. By assumption, we may take N0,N1,N2 ∈K, that can not be amal- gamated. +> For ∈ 2 define a family of M ∈K so that the following hold:

1. |M| = (1 + `()), 2. l → M K M, S 3. when `() is a limit ordinal, M = <`() M, 4. Mˆ0 and Mˆ1 cannot be amalgamated over M. Using -categoricity andS the triple N0,N1,N2 the construction is possible. ∈ +> For 2 let M := <+ M. Now suppose that M ∈K+ is universal. Without loss of generality we may + assume that |M| = +. By universality for every ∈ 2 there is a K-embedding f : M → M. Now consider C := {<+ | = (1 + )}, it contains a club. Using + + there are =6 ∈ 2 and ∈ C as in + . Denote by the largest common initial segment of and (it is ). Since [] =6 [] we assume that []=0and []=1. Pick M K M of cardinality containing the set f[Mˆ0 ] ∪ f[Mˆ1 ]. Note that the diagram

/ Mˆ0 MO O f

id f / M Mˆ1 id is commutative in contradiction to requirement 4 in the construction.

The argument used in the proof of Theorem 4.6 can be used to prove the fol- lowing useful: CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 19

+ COROLLARY 4.9. Suppose 2 < 2 and K is an AEC with LS(K) . If I(, K)=1and there exists >such that I(, K)=1then K has the -amalgamation property. K3 5. Solution to Baldwin’s question and | ℵ ∈ Suppose T L(Q).GivenM = T of cardinality 1 pick a Lω1,ω a Scott sentence of a countable substructure of M. Now letVLA Lω1,ω(Q) be a countable fragment containing T and . Let K := Mod( T ∧ ) and let N1 K N2 iff N1 TV,LA N2. It is enough to prove: hK i K THEOREM 5.1. Suppose , K is an AEC which is PCℵ0 .If is categori- ℵ ℵ K 6 ∅ cal both in 0 and 1 then ℵ2 = .

PROOF. Since K is categorical in ℵ1 and closed under union it is enough to show that 6 ∈K ( ) There are N0 = N1 ℵ1 such that N0 K N1. The following concept is central to the theory: DEFINITION 5.2. K3 { ∈K := (M,N,a):M,N ,M K N, a ∈|N||M|}. K3 On define a partial ordering by 0 0 0 def 0 (M,N,a) < (M ,N ,a) ⇐⇒ M K M ∧ 0 0 0 M =6 M ∧ N K N ∧ a = a . K K 6 ∅ K3 6 ∅ When is categorical in then the assumption + = implies that = . REMARK 5.3. In the next section we will show that the element (M,N,a) of K3 plays a similar role to that of tp(a/M, N) in first-order logic.

Using () and the assumption that I(ℵ1, K)=1once more, Theorem 5.1 follows from: hK i K THEOREM 5.4. Suppose , K is an AEC which is PCℵ0 .If is categor- ℵ hK3 ,

B |= n+1 <n for all n<ω. The main step is the following: hK i LEMMA 5.6 (technical lemma). Suppose , K is an AEC which is PCℵ0 , K ℵ ℵ ∈K and is categorical both in 0 and 1. Then for every M ℵ0 there are { ∈K } Mn T ℵ0 n<ω such that for every n<ωwe have that Mn+1 K Mn and M = n<ω Mn.

PROOF. By categoricity in ℵ0 it is sufficient to show that there exists a model M as in the statement. Let T1, 1,T2, 2 be at most countable such that K = PC(T1, 1,L(K)) and {h|N|, |M|i : M K N} = PC(T2, 2, {P (x)}). hK i K 6 ∅ Since , K is an abstract elementary class and ω1 = we can fix an K- { ∈K } increasing continuous chain of models M Sℵ0 : <ω1 . Denote by f : ω →Kℵ the mapping 7→ M . Let M := M . For every a ∈|M| 1 0 <ω1 let h(a):=Min{<ω1 : a ∈|M|}. By the reflection principle there exists a regular cardinal sufficiently large so that + H() { ,f,h,M,T1, 1,T2, 2,L(K), ...} and hH(), ∈i reflects all relevant information. Namely

hH(), ∈i |=∀a ∈|M|[h(a) ∈ ω1] ∧∀<∈ ω1

(1) [f() ∈ PC(T1, 1,L(K)) ∧

hf(),f()i∈PC(T2, 2, {P (x)})]

(2) hH(), ∈i |= ∀<∈ ω1[<→ f() K f()]

hH(), ∈i |=∀ ∈ ω1[∀[<→ +1<]] → (3) ∀M ∈ Rang(f)[M K f() ∧ M =6 f()] → ∃i<[M K f(i)]]

(4) hH(), ∈i |= “hK, Ki is an AEC”

(5) hH(), ∈i |= “hK, Ki satisfies Theorem 1.12” [ hH(), ∈i |= M = f()

<ω1 (6) hH(), ∈i |= ∀a ∈|M|[a ∈ f(h(a))]

∧ (∀ ∈ ω1)[

Let h ∈ | | i A := H(), , M ,ω,f,h,Q,T1, T21, 2,ϕ,p, <,ϕ∈T1∪T2,p∈1∪2 .

Where 1, 2 are unary predicates interpreted by the corresponding sets of types, similarly Tl are unary predicates interpreting Tl, Q is a unary predicate interpreted by the set of ordinals ω1, ω is a unary predicate interpreted by the set of natural numbers and f and h are unary function symbols interpreted by the corresponding functions. ,ϕ and p are individual constants interpreted by the corresponding elements. Now let p1(y):={1(y) ∧ y =6 q : q ∈ 1}, p2(y):={2(y) ∧ y =6 q : q ∈ 2} and p3(ϕ):={T1(ϕ) ∧ ϕ =6 : ∈ T1}, p4(ϕ):={T2(ϕ) ∧ ϕ =6 : ∈ T2}, finally let p5(j):={(j) ∧ j =6 : <}. Denote by T3 the theory of A. Clearly A ∈ EC(T3, {p1,p2,p3,p4,p5}) and ∈ { } B B for all B EC(T3, p1,p2,p3,p4,p5 ) since Tl = Tl, l =l and we have (using ()0 and ()1) that B For all <∈ Q [f() K f()] ∧ [f() ∈ PC(T1, 1,L(K))]. B KB K Since B omits p5 we get that = , namely we have that / = = / =. Since hQA, ∈i has order type + and from the assumption on an application of Fact 5.5 to EC(T3, {p1,p2,p3,p4,p5}) produces a model B B ∈ EC(T3, {p1,p2,p3,p4,p5}) such that there exists {n : n<ω}Q such that for every n<ωwe have that B |= n+1 <n.Forn<ωlet Mn := f(n). We conclude with showing that T ∈KB CLAIM 5.7. There exists N such that N = n<ω Mn. PROOF. Since ω QB and for all k<ωand for every n<ωwe have

B |=[k<n], B the set I := { ∈ Q : ∀n<ω[<n]} is nonempty and directed. Since B ShK, Ki is an abstract elementary class (by ()3) we have that there exists N := s∈I f(s). ByT the definition of N we have that for all n<ωN K Mn.So clearly N n<ω Mn. Using the functionT h we show that the last two sets are ∈ | | equal. Suppose that there exists a n<ω Mn N .By( )5 there exists a first ∈ Q such that a ∈ f(). Since a ∈ Mn for every n by minimality we get that n for all n<ω. Since f is order-preserving we get that f() K f(n), namely f() Mn for all n which is a contradiction to the choice of the element a. (M,N,a) ∈K3 S ω {M S : Suppose ℵ0 is maximal. Given any 1, define }K <ω1 ℵ0 as follows: | S| 1. M = ω(1 + ), S S S 2. For limit, M = < M , 3. for = +1, there are two cases: ∈ ℵ S S (a) if S, using 0-categoricity take M+1 K M and S ∈| S || S| S S S a M+1 M so that (M ,M+1,a ) = (M,N,a) 22 RAMI GROSSBERG

(b) if 6∈ S, then take a descending decomposition. I. e. apply Lemma S 5.6 to the model M to get a descending {M | n<ω}Kℵ such T n 0 S S that M = n<ω Mn. Now define M+1 to be M0. S S < =⇒ M K M Notice that S . For each set S let M := M S. S <ω1

6 S1 6 S2 CLAIM 5.8. If S1 S2 mod Dω1 then M = M .

6 S1 S2 PROOF. Let S1 S2 mod Dω1 and f : M = M be given. By require- S S ment (1) of the construction |M 1 | = |M 2 | = ω1. Observe that

C1 := {<ω1 :(∀<)[f() <]} is a club. Using continuity of the chains we get that { S1 S2 } C2 := <ω1 : f : M = M is also a club. Take C := C1 ∩ C2. Since C is a club we may assume without loss of generality that there exists ∈ C ∩ (S1 S2). Since ∈ S1 by the construction we have that (M S1 ,MS1 ,aS1 ) K3 +1 is a maximal element of ℵ0 . Since S1 6∈ S2 f(a ) M , by the assumption 6∈ S2 there exists n<ωsuch that S1 6∈ f(a ) Mn. 1 S2 Let N := f [Mn]. Since M is a proper substructure of Mn we have that S1 ( | | | S1 | M N. Since N is a countable subset of M there exists <ω1 such that S1 ∩ ∈ ∩ N M . Since C (S1 S2) is unbounded there exists C (S1 S2) max{, } (N,MS1 ,aS1 ) K3 greater than . Thus we have is a proper ℵ0 -extension S1 S1 S1 of (M ,M+1,a ) contradicting its maximality. ℵ By Ulam’s theorem the claim gives I(ℵ1, K)=2 1 .

6. Galois types Recall the following basic result of elementary model theory:

FACT 6.1. Let C be an uncountable saturated model of cardinality greater than |L(C)|.Fora, b ∈|C | and A |C | such that |A| < k C k.

tp(a/A) = tp(b/A) ⇐⇒ ∃ f ∈ AutA(C) such that f(a)=b.

Namely the orbit of the element a under the group action of AutA(C) on C can be identified with tp(a/A). The set of L(K)-formulas satisfied by a does not have the corresponding property for abstract elementary classes. Thus we need a replacement. A replacement introduced by Shelah in [Sh 300] and since takes a prominent role in model theory of AECs is to work directly with orbits instead of set of formulas! This is the notion of Galois type to be defined below. Unfortu- nately not having formulas (in any logic) creates many technical difficulties. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 23

DEFINITION 6.2. Let M ∈K, We say that (M,N1,a1) and (M,N2,a2) ∈ K3 ,are -related, written

(M,N1,a1) (M,N2,a2), if there exists N ∈K and K-embeddings

h1 : N1 → N and h2 : N2 → N, such that

h1(a1)=h2(a2) and the following diagram commutes:

h1 / NO1 NO

id h2 / M N2 id

LEMMA 6.3. Suppose that K has the amalgamation property. For M ∈K K3 the relation is an equivalence relation on .

PROOF. Exercise, notice that we use that M and N` are amalgamation bases.

DEFINITION 6.4 (Galois types). ∈K3 1. For (M,N,a) ,welet ga-tp(a/M, N)=(M,N,a)/ . This is the type of a over M in N. 2. For M ∈K,welet { 0 0 0 0 ∈K3 } ga-S(M)= ga-tp(a /M, N ):(M,N ,a) .

3. Given p ∈ ga-S(M) and N ∈K, we say that p is realized by a ∈ N,if 0 there exists N K N of cardinality containing |M|∪a such that 0 ∈K3 0 (M,N ,a) and p = ga-tp(a/M, N ). K Assuming that ℵ0 has the amalgamation property, the content of Theorem 5.4 is a weak replacement of the compactness theorem: K THEOREM 6.5 (extension property of types). Let be an AEC which is PCℵ0 . ℵ K ∈K Then the assumption I( 1, )=1implies that for every M K N ℵ0 and every p ∈ ga-S(M) there exists q ∈ ga-S(N) extending p. ℵ K Thus the assumptions of categoricity in 1 together with ℵ0 has the amalga- mation property are a replacement of an easy fact in first-order logic which is a corollary of the compactness theorem. This is a typical example of “compactness regained” which appears also in much more complicated results.

DEFINITION 6.6. Let hK, Ki be an abstract elementary class and suppose that >LS(K).ForN ∈K the model N is -Galois saturated iff for every M K N of cardinality less than and every p ∈ ga-S(M) is realized in N. 24 RAMI GROSSBERG

The contents of the following theorem that for AEC classes that have the amal- gamation property, model homogeneity and saturation are equivalent properties. It is further evidence that the notion of Galois type makes sense.

THEOREM 6.7. Let hK, Ki be an abstract elementary class and suppose that >LS(K). Suppose that K has the (<,<)-amalgamation property then for M ∈K we have that M is -Galois saturated iff M is -model homogeneous.

PROOF. It is easy to show that homogeneity implies saturation. Let LS(K) be such that <. Let N1 N2 ∈K be given such that N1 M. We may assume that is the cardinality of N2. Fix hai | i<i an enumeration of N2. Now by induction on i<define two increasing continuous chains of h i | ∈{ }i h | i models Nl ,fi i , l 1, 2 and mappings fi i< satisfying: 0 k ik 1. Nl = Nl,f0 = idN1 , N2 = , i i 2. N1 K N2, i → 3. fi : N1 , M and ∈ i+1 4. ai N1 for every i<. Since the chains are continuous we only have to define two models and an embedding for i = j +1: j j j j j Let M1 := fj[N1 ]. Using Remark 1.6 let M2 be an amalgam of N2 and M1 j j j j j over N1 such that M2 M1 and let gj : N2 = M2 be an extension of fj. Namely the diagram

g j j / j NO2 MO2

id id

N j /M j /M 1 fj 1 id commutes. ∈ j If gj(aj) M1 then do nothing. j j Otherwise consider p := ga-tp(gj(aj),M1 ,M2 ) and use the hypothesis that + ∈K ∈ j M is 2 -saturated to get M 2 and to find b M such that M1 M M j j j and (M1 ,M2 ,gj(aj)) (M1 ,M ,b). Unwinding the definition of gives: There exists N ∈K and mappings h1,h2 such that the diagram

j h2 / MO2 NO

id h1

M j /M 1 id CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 25 commutes and in addition we have h2(gj(aj)) = h1(b). By renaming the elements of N and changing the mapping h1 accordingly we may assume that h2 is the identity. Thus by gluing the last two diagrams together we get that the diagram

g j j / j id / NO2 MO2 NO

id id h1

N j /M j /M /M 1 fj 1 id id i j i commute. Now pick N2 N2 and hj gj such that hj : N2 = N . So we have that i NO2 MM MMM MMhMj id MMM MM& N j /N O2 gj O

id h1

N j /M 1 fj

i 1 Commutes. Let N1 := hj [h1[M ]]. ∈ ∈ j+1 Since gj(aj)=h1(b) (using also that b M ) we get that aj N1 and from hj gj and the fact that i N2 NN O NNN NNhNj NNN NN' id NO

h1

N j /M 1 fj i j commutes we get that N1 N1 . Now we are ready to define the mapping fi, let 1 j K fi := h1 (hj N1S). Itisa -embedding that extends fj as required. Verify using Axiom A2 that i< fi is an embedding of N2 over N1 into M.

7. Stability-like properties K3 REMARK 7.1. Notice that has no maximal element corresponds to “every 0 0 type over M can be extended to a type over M for all M ( M ∈K”. The K ℵ contents of Theorem 5.4 is that if is an PCℵ0 class categorical in 0 such that ℵ I(ℵ1, K) < 2 1 every type over a countable model has a proper extension. 26 RAMI GROSSBERG

THEOREM 7.2 ([Sh 576]). Suppose K is a PC-class. Let . K + K K 6 ∅ 3 If I(, )=I( , )=1and ++ = , then K has no maximal element. ∈K3 DEFINITION 7.3. (M0,M1,a) is minimal iff for every 0 0 (M0,M`,a`) h` (M0,M1,a) ` =1, 2 0 0 we have that ga-tp(a1,M0,M1) = ga-tp(a2,M0,M2). 0 0 0 0 (M0,M1,a) h(M0,M1,a) stands for “h is an embedding of M1 into M1 taking 0 0 M0 into M0 and h(a)=a ”. The above notion of minimal triple is a generalization of strongly minimal non algebraic type. We are actually generalizing the stationarity property of strongnly | 0 | minimal types. I.e. if a1,a2 realize extensions (to M0 ) of the minimal type of a then they types of a1 and a2 are equal. There are several basic existence results of strongly minimal types. I.e.

FACT 7.4. (W. Marsh) If T is an ℵ0-stable first-order then there exists a strongly minimal formula. It is natural at this stage to introduce the following “obvious” concept:

DEFINITION 7.5. K is stable in iff | ga-S(M)| for every M ∈K . An analog to Marsh’e theorem is the following result:

THEOREM 7.6. Let LS(K). Suppose K has the -amalgamation property K3 ∈K3 and does not have a maximal triple. If there is no minimal (M0,M1,a) , let p := ga-tp(a/M0,M1) then the following holds: 0 ∈K 0 1. There is M0 such that M0 M0 and 0 0 log() |{p ∈ ga-S(M0):p p}| 2 ; ∈K 0 2. There exists N + such that M0 N and |{ ∈ 0 }| + p ga-S(M0):p is realized in N = . Namely, |{ ∈ ∃ 0 0 0 0 ∈K c N : N N,M0 N ,N , such that 0 0 }| + ga-tp(c/M0,N ) ga-tp(a/M0,M1) = . The previous theorem is further evidence that the notion of Galois types is useful. K3 The notion of minimal element of plays a central role in proving categoric- ity results. A key to categoricity: Under strong assumptions on K we have that if p ∈ ga-S(M) is minimal then for every M2 M1 M if M1 ( M2 then there exists a ∈|M2||M1| realizing p. At the current state of affairs there is no nice forking-like relation for AECs (even under the assumption that they are categorical above the hanf number). However there are several approximations. Since not even the parallel to Mor- ley’s theorem is available for AECs one can investigate one of the coarser notions from the days of stability theory before forking. Below it is shown that a notion parallel to splitting of types is moderately nicely behaved. For this we make the following CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 27

HYPOTHESIS 7.7. K has the (, < , )-amalgamation property for every , and , we may assume that there exists a monster model C.Bymonster model we mean a very large model homogeneous model. Notice that existence of large Galois-saturated models is also desireable, but we need more than just amalgamation to prove their existence.

DEFINITION 7.8. Suppose N K M ∈K, kNk a cardinal number. The type p ∈ ga-S(M) -splits over N iff there exists N ,N ,hof cardinality such 1 2 that N N` M, h : N1 = N2 and the types p N2 and h(p N1) are N contradictory (= there if N is an extension of M then there is no a ∈|N | such that a |= p N2 ∪ h(p N1)). It is tempting to call the previoous notion Galois-splitting. However I feel that doing so will make certain passages unreadable. It is important to recognize the similarity of Galois-splitting to the ususal first-order splitting as well as the differences. The key difference that here we don’t have a formula wittnessing the splititng, moreover the splitting is evidenced by models rather than a finite sequence of parameters.

THEOREM 7.9. Let LS(K). Suppose K is stable in . For every M ∈ K and every p ∈ ga-S(M) there exists M0 M of cardinality such that p does not split over M0.

PROOF. Suppose N K M, a ∈ N such that p = ga-tp(a/M, N) and p splits over N0, for every N0 K M of cardinality . Let := min{ | 2 >}. Notice that and 2< . We’ll define {M M | <}K increasing and continuous K-chain ∈K which will be used to construct M such that | | ga-S(M) 2 >obtaining a contradiction to -stability.

Pick M0 M any model of cardinality . For = +1; since p splits over M there are N K M of cardinality ,` for ` =1, 2 and there is h : N,1 =M N,2 such that h(p N,1) =6 p N,2. Pick M K M of cardinality containing the set |N,1|∪|N,2|. ∈K ∈ K Now for <define M and for 2 define a -embedding h such that ⇒ 1. < = M KSM, 2. for limit let M = < M , 3. <∧ ∈ 2=⇒ h h, K ∈ ⇒ → 4. 2= h : M , M and 5. = +1∧ ∈ 2=⇒ hˆ0 (N,1)=hˆ1 (N,2). The construction is possible by using the -amalgamation property at = +1several times. Given ∈ 2 let N be of cardinality and f0 be such that 28 RAMI GROSSBERG the diagram

f0 / MO+1 NO

id id / M M h commutes. Denote by N2 the model f0(N,2). Since h : N,1 =M N,2 there is a K-mapping g fixing M such that g(N,1)=N2. Using the amalgamation property now pick N ∈K and a mapping f1 such that the diagram

f1 / MO+1 NO

id id N / ,O1 g NO2

id id / M M h ∈K Finally apply the amalgamation property to find M+1 and mappings e0,e1 such that e1 /M NO O+1

id e0

/ M N id commutes. After renaming some of the elements of M+1 and changing e1 we may assume that e0 =idN . Let hˆ0 := f0 and hˆ1 := e1 f1. Now for ∈ 2 let [ [ M := M and H := h. < < K Take N K M from , an amalgam of N and M over M such that

H /N NO O

id id / M M h commutes. Notice that 6 ∈ ⇒ 6 = 2= ga-tp(H(a)/M,N ) = ga-tp(H(a)/M,N ). CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 29

| | Thus ga-S(M) 2 >.

QUESTION 7.10. Is it possible to replace ℵ0 in Remark 7.1 with an uncount- able ? This is important since a positive answer will require developing a replacement for the undefinability of well ordering in Lω1,ω. The fact that undefinability of well ordering does not have a natural generalization for uncountable cardinals follows from an independence result of Jon Barwise and Ken Kunen [BaKu]. Here is a very simple example of a result from [Sh 576] in this direction:

THEOREM 7.11. Let . Suppose K is an AEC such that K is PC, K + K K 6 ∅ K3 I(, )=I( , )=1and ++ = . Then does not have a maximal triple. K3 K K 6 PROOF.( has no max element). Since I(, )=1and + =0,itis enough to show that ∈K3 if (M0,M1,a) , then there exists 0 0 0 (M0,M1,a) (M0,M1,a) 6 0 such that M0 = M0. ∈K3 Suppose for contradiction that (M0,M1,a) is a maximal triple. + Define {Ni : i< }K increasing continuous + and {hi : M1 = Ni+1 : i< } such that Ni = hi[M0]. Since we want these sets to be continuous, it is enough to define them at successor stages. Given Ni, + where i< ,byI(, K)=1there exists an isomorphism h : M0 = Ni. Since M K M , we can find N and h such that h h and 0 1 i+1 i i h : M1 = Ni+1. Note that hi(a) ∈ Ni+1 Ni. From the construction we get that ∈K3 + that (Ni,Ni+1,hi(Sa)) is a maximal triple, for i< . Now let N = i<+ Ni. + + Then, since hi(a) ∈ Ni+1 Ni for all i< and since {Ni : i< } is a chain, + 1 N ∈K+ . Since I( , K)=1and K++ =06 , there is N =6 N in K+ such that 1 N K N . { 1 1 +}K Pick N K N : < increasing and continuous such that for + 1 6 1 | 1|6 | | all < ,N N but N = N . (This is possible since N = N ). Define a function g : |N 1|→+ as follows: ( i if h (a)=b g(b):= i 0 Otherwise. Notice that the relation g(b)=i is a function since we have seen above that i =6 j =⇒ hi(a) =6 hj(a). Applying the reflection theorem from set theory, let be large enough such that V contains the set { 1 h 1 +i + N,N ,M0,M1,a, Ni,Ni ,hi : i< ,g, , “the PC definition of K ”} and such that the model B h 1 K 7→ 1 i = V,,N,N ,M0,M1,Q, ,g,i (Ni,Ni ,hi),a , 30 RAMI GROSSBERG where Q = +, reflects the following sentences: (i) ”{Ni : i<} is increasing and continuous in K+” ∀ ∈ 1 ∧ 6 1 (ii) ( i Q)[Ni Ni Ni = Ni ] ∈ K3 (iii) ”for all i Q, (Ni,Ni+1,hi(a)) is a maximal element in ” (iv) g : |N 1|→Q 1 (v) (∀b ∈ SN )(∃i ∈ Q)((hi(a)=b) → g(b)=i) (vi) N = Si∈Q Ni 1 1 (vii) N = i∈Q Ni . B Let BK B of cardinality such that Q = , a limit ordinal. (For example, take {Bi K B : i< } increasing continuous such that ||Bi|| = and B B i { + } Q i. Then useS the fact thatS < : = Q is a club.) B 1B 1 Now, N = < N = < N = N. Similarly, N = N . Denote a := h(a). ∩ 1 B 6∈ 1 Claim N+1 (N ) = N. In particular, a N . ∈ 1 ∈B B ∈B PROOF.Ifa N then a . Since is closed under g, g(a) . But recall that B|=(g : |N 1|→Q) ∧ (∀b ∈ N 1)

((∃i ∈ Q)(hi = b) → (g(b)=i)).

So g(a) ∈ Q. Compute g(a)=. Since B|= ZF, also +1∈ QB, which contradicts the fact that = QB. ∩ 1 If N+1 N properly contains N, then since ∈ 1 6∈ 1 1 1 ∈K3 a N+1 N+1, by the fact that a N we get that (N ,N+1,a) . And since then 1 1 (N,N+1,a) < (N ,N+1,a), we get a contradiction to the assumption that (N,N+1,a) is a maximal triple.

In [GrVa] Grossberg and VanDieren have shown that for categorical AEC Morley-sequences exist when the dependence relation is non splitting. univ Denote by M K N the statement N is universal over M i.e. for every M 0 M of cardinality kMk there exists a K-embedding from M 0 into N over M.

DEFINITION 7.12 (from [GrVa]). Let K be an AEC. The class is called - superstable iff there exists >LS(K) satisfying univ 0 1. for every M ∈K, there exists M K M ∈K and + univ 2. for every =cf() < whenever hMi ∈K | i i is K - increasing and continuous and p ∈ ga-S(M), there exists i<such that p does not -split over Mi.

THEOREM 7.13 (from [GrVa]). Suppose K is -superstable for some LS(K) and K has the amalgamation property. Let M ∈K>, A, I M be CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 31 given such that |I|+ > |A|. Then there exists J I of cardinality +, indiscernible over A. Moreover J can be chosen to be a Morley sequence over A. Where + DEFINITION 7.14 ([GrVa]). {ai ∈ } is a Morley sequence over M0 iff + there exists an K-chain {Mi | i< } such that pi := ga-tp(ai/Mi,Mi+1) does not -split over M0 and hai | i ∈ i is an indiscernible sequence over M0.

8. proof of Theorem 4.3 Before starting the proof let me point out that we will be using eth Devlin- Shelah weak diamond. Instead of using the principle + we use a principle that may look little stronger but using a pairing function together with with Fact 4.7 one can show that it follows from + : We will be using: There exists a stationary subset S of + such that for every + + + F : < + < + < + → 2, there exists a guess g : + → 2, such that for every , , h : + → +, the set { ∈ S | F ( , , h )=g()} is stationary.

PROOF OF THEORM 4.3. Recall that I(, K)=1and K fails to have the - amalgamation property implies that K+ is nonempty. By assumption, we may take N0,N1,N2 ∈K, that can not be amalgamated. +> For ∈ 2 define a family of M ∈K so that the following hold: 1. |M| = (1 + `()), 2. l → M K M, S 3. when `() is a limit ordinal, M = <`() M, 4. Mˆ0 and Mˆ1 cannot be amalgamated over M. Using -categoricity and the triple N0,N1,N2 the construction is possible. Divide the proof into two cases, in the first case assume a stronger failure of the amalgamation and the second is the negation of the first. 0 Case A: Suppose that there exist N M ∈K so that for every M extending 0 1 0 0 1 M in K, there is a pair M and M extending M so that M and M cannot be amalgamated over N. To requirements 1-4 we add Mhi = N and replace (4) by

0 (4) Mˆ0 and Mˆ1 cannotS be amalgamated over N. ∈ + For 2, let M := <+ M. 6 ⇒h i 6 h i CLAIM 8.1. = = M,a a∈|N| = M,a a∈|N|. PROOF. Let be the meet of and . If there was an isomorphism between h i h i M,a a∈|N| and M,a a∈|N|, we would have that M is an amalgam of Mˆ0 and Mˆ1 over N, a contradiction to requirement (4). 32 RAMI GROSSBERG

Suppose for the sake of contradiction that

+ := I(+, K) < 2 .

Take {Mi | i<} to be a complete set of representatives for K+ . Clearly,

+ |{hM,aia∈|N|/ = : ∈ 2}|

|{hMi,aia∈|N| : i<}| + + kMik = ( ) = 2 = + 2 < 2 .

But this is a contradiction to the fact we have many pairwise non-embeddable + models. Notice how we used the assumption 2 < 2 in the final step. 0 Case B: Suppose for all N and every M ∈K there is an M extending M so 0 that for any extensions M0 and M1 of M , M0 and M1 can be amalgamated over N. Again, we tweak requirement (4) and replace it by:

0 0 1 0 1 (4) if M and M are extensions of Mˆ0 and Mˆ1 , then M, M and M can be amalgamated. To perform the construction at successors, we do the following, to define Mˆ0 and M from M : ˆ1 Apply categoricity to fix an isomorphism f : M = N0 (the unamalgamable triple we chose at the beginning). Using N1,N2 and their preimages pick M and M extensions of M which cannot be amalgamated over M. By the assumption of case B, we can take Mˆ0 to be an extension of M so that any extensions of Mˆ0 can be amalgamated over M. Take Mˆ1 similarly for M . Let C := {< : = (1 + )}, notice that it is a club. By Ulam’s theorem + there are {S C : < } stationary sets such that 6 ⇒ ∩ ∅ + 1 = 2 = S1 S2 = and for all < we have that + (S) holds. For every <+ such that = (1 + ), h : → and , ∈ 2 let  1 if h : M ,→ M and    MOˆ0    id F (, , h):= M /  Mˆ0  h   can be amalgamated.    0 Otherwise. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 33

+ By + (S) pick g : → 2 such that + for all , ∈ 2 and every h : + → + we have that 0 { ∈ } S := S : F ( , , h )=g() is stationary. For X + and every <+ let  g() if ∈ S and ∈ X X []:=  0 Otherwise.

+ Notice that since {S : < } are pairwise disjoint for any ∈ X there is at most one such that ∈ S (maybe none), so X is well defined. We finish by showing: 6 + 6 CLAIM 8.2. For every X = Y we have that MX = MY . + PROOF. Suppose X =6 Y , h : M ,→ M . An application of 0 X Y + (S) to X ,Y and h yields S as above. { + | → } + 00 0 ∩ Let D := < (h ): be a club. For < let S := S D. ∈ ∈ 00 Without loss of generality there exists X Y . Pick S . Denote by the sequence X and by the sequence Y . Since /∈ Y by the definition of the sequence Y we have that Y []=0, namely l ˆ0 l Y . Now consider X []. There are two possibilities (according to the value of X []):

1. If X []=1, by the definition of X [] necessarily X []=g(), since ∈ 0 S we have that F (, , h)=1. From the definition of F we get that 1 Mˆ0 and Mˆ0 can be amalgamated over M. Denote by M the amalgam and let f and g be such that the following diagram commutes:

M / 1 ( )1 Oˆ0 g MO

id f / M Mˆ0 h l l → Since ˆ1 X , ˆ0 Y and from the assumption that h : MX ,

MY , the following diagram / Mˆ1 M O h OY id / M Mˆ0 h

2 must commute. By axiom A4 there exists M K MY of cardinality 2 2 such that h(Mˆ1 ) K M and Mˆ0 K M . 34 RAMI GROSSBERG

Namely the following diagram commutes: / ()2 Mˆ1 2 O h MO id / M Mˆ0 h Since

MO1 f

/ 2 Mˆ0 id M 0 3 by the second half of requirement (4) there are M ∈ K and K-mappings el : M l ,→ M 3 (for l =1, 2) such that

/ ()3 1 3 MO e1 MO

f e2 / M 2 id M is commutative. Combining ()1, ()2 and ()3 together we get

/ / ()4 Mˆ0 1 3 O g MO e1 MO

id f e2 / / M M m6M 2 h midmm mmm mmm id mmm  mmm h Mˆ1 Thus we have that he 2 / 3 MOˆ1 MO

id ge1 / M Mˆ0 id is commutative, which is a contradiction to the first half of requirement (4)0.

l → 2. If X []=0then we have that ˆ0 X . Since h : MX , MY we get

that MY is an amalgam of Mˆ0 and Mˆ0 over M so by the definition of ∈ 00 F we get that F (, , h )=1. Since S we have that g()=1 which by the definition of X gives X []=1and this contradicts the assumption of this case. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 35

9. The main problems In my opinion most of the future results will come from two separate lines of research, that many years from now will merge. 1. Stability theory for AECs. The most ambitious form of this program can be stated as follows: Let MT stand for Morley’s categoricity theorem, f.o. stability for the contents of Shelah’s book ([Sh c]) and let n.e.categ stand for the partial results toward Shelah’s categoricity conjecture (the contents of [Ke2],[Sh48], [Sh87a], [Sh87b], [MakSh], [KoSh], [Sh 394], [Sh472], [Le1]). By developing stability theory for AECs I mean solving the equation f.o. stability x () = . MT n.e.categ Examples of progress toward this direction can be found in [Sh3], [Sh88], [Sh 300], [GrHa], [Gr3],[Gr4], [GrHa],[GrLe1],[GrLe2], [GrLe4], [GrSh1], [GrSh2], [GrSh3], [GrSh4],[Le1], [Le2],[Sh 576], [Sh 600],[ShVi], [GrVa],[Kov1] and [Va].

2. Geometric model theory for AECs. This program can be described as an attempt to solve the following equation: HZ ? (#) = f.o. stability x where HZ stands for Hrushovski’s extension of Zilber’s geometric ideas, and x is a (partial) solution for (). There very few results in this direction. Among them are [Le1], [GrLe3] and [Le3].

I expect that several interactions of () and (#) will eventually yield among other things a solution for Shelah’s categoricity conjecture.

While I am convinced that eventually the theory will have more applications to main stream mathematics via commutative algebra, algebraic geometry or analytic structures than model theory of first-order logic, so far there are no applications in sight. The greater potential is due to the ability to axiomatize local finiteness and structures satisfying various chain conditions. It is too early to predict what exactly these applications will be. It is natural to expect that studying AECs of some concrete structures (rings and groups) may produce valuable results. There is one conjecture that may eventually be solved using non elementary methods. This is Zilber’s conjecture (stronger than Schanuel’s conjecture) con- cerning analytic structure from [Zi]. Namely that Cexp is the canonical structure ℵ of cardinality 2 0 in the class H(ex/st). 36 RAMI GROSSBERG

The following are major concrete problems in AEC: 1. In Makkai-Shelah [MakSh] as well as in [Sh 394] and [Sh472] partial re- sults toward a categoricity theorem are presented. In both cases the hypoth- esis is that the class is categorical in a successor cardinal. It is not clear at all if this is just a technical limitation or a central problem. I suspect it is central. Probably replacing the assumption that the class is categorical in + with the assumption of categoricity in is of similar difficulty. 2. The categoricity theorem in [Sh 394] is a going down theorem. Is there a going up theorem? The simplest instance of this is:

CONJECTURE 9.1. Let K be an AEC. If there exists Hanf(K) such that K is categorical in then K is categorical in +. 3. It is a major open problem to find a nice (forking-like) dependence notion for AEC. In fact even under the assumption that K has the amalgamation property and the class is categorical in a cardinal above Hanf(K) this is open (see Remark 4.10(1) in [Sh 394]). 4. One of the technical problems of working in AEC without the amalgama- tion property is the inexistence of monster models and therefore types over models can not always be extended to global types. To deal with this Shelah and Andres Villaveces have introduced in ([ShVi]) an interpolant, which is the framework of AEC without maximal models, under GCH. They have managed to show that categoricity implies that every small model (below the categoricity cardinal) can be extended to an amalgamation base and sev- eral other basic facts. Some extensions of this work can be found in [Va]. As of today there are no categoricity results in this context. It is natural to expect the following:

CONJECTURE 9.2. Let K be an AEC without maximal models and sup- pose that Hanf(K).IfK is categorical in + then K is categorical in every . 5. Probably the appropriate name for this is existence of Hanf number for amalgamation:

CONJECTURE 9.3. Let K be an AEC. Suppose (for simplicity?) that K does not have maximal models. There exists a cardinal number (K) such that if K has the (K)-amalgamation property then K has the - amalgamation property for all (K).

References [BaLe] John T. Baldwin and Olivier Lessmann. Finite models, to appear. [BS1] John T. Baldwin and Saharon Shelah, The primal framework. I. Annals of Pure and Applied Logic, 46:235–264, 1990. [BS2] John T. Baldwin and Saharon Shelah, The primal framework. II. Smoothness. Annals of Pure and Applied Logic, 55:1–34, 1991. [BS3] John T. Baldwin and Saharon Shelah, The primal framework. I. Annals of Pure and Applied Logic, 46:235–264, 1990. CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 37

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